Difference between revisions of "Chakravala method"
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Revision as of 22:03, 2 March 2023
The chakravala method is an algorithm for solving the Pell equation
Contents
Method of composition
We let and
be integers such that
, and we notate
.
We then choose an integer and let
Existence of suitable choice
We claim that it is always possible to choose such that
is an integer.
Because , we have
, so
Suppose . Then
Because
,
also divides
, so
.
We can construct a set of possible integer values of
, none congruent to another
; the corresponding values of
take all
distinct values
, so there must be one element
in the set such that
; that is,
is an integer.
Recovery of initial conditions
We further claim that if is an integer, then
is also an integer, and
.
Evaluation
We now claim that .